Hida theory for special orders

TL;DR

This paper develops a Hida theory framework for families of quaternionic modular forms associated with special orders, extending classical results to a setting with 2-dimensional Hecke-eigenspaces.

Contribution

It establishes a Control Theorem for these families, interpolating 2-dimensional Hecke-eigenspaces in the context of quaternionic modular forms from Pizer's orders.

Findings

Proves a Control Theorem for quaternionic modular forms

Interpolates 2-dimensional Hecke-eigenspaces

Extends classical Hida theory to special orders

Abstract

This note is devoted to the study of families of quaternionic modular forms arising from orders defined by Pizer. In this situation, the Hecke-eigenspaces are 2-dimensional contrary to the classical case of Eichler orders. The main result is a Control Theorem in the spirit of Hida, interpolating these 2-dimensional Hecke-eigenspaces. We restrict our attention to a definite rational quaternion algebra ramified at a single odd prime $\ell$.